Absorbing boundary conditions for numerical simulation of waves

Engquist, Björn; Majda, Andrew J.

doi:10.1073/pnas.74.5.1765

Cited by 523 publications

(394 citation statements)

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“…Note that this boundary condition assumes a plane wave propagating in the positive z-direction E r ∝ exp(−iβz). This boundary condition is described in detail in [38,41,42].…”

Section: Homogeneous Inhomoneous Dirichletmentioning

confidence: 99%

“…This BC is often referred to as the absorbing BC (see [38]), however since we use absorption in another meaning and since this BC deals with the propagation of the wave outside the domain we prefer to call this the Propagation BC assume that incident power driven into the plasma has a TEM mode so that the corresponding E-field component reads…”

Section: Homogeneous Inhomoneous Dirichletmentioning

confidence: 99%

See 1 more Smart Citation

Effect of remote field electromagnetic boundary conditions on microwave-induced plasma torches

Jimenez-Diaz¹,

Dijk²,

Mullen³

2011

J. Phys. D: Appl. Phys.

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Abstract.A flexible versatile electromagnetic model constructed with the PLASIMO platform is employed to explore electromagnetic features of microwave induced plasma torches. The bases, formed by a full-vector formulation of the Maxwell equations, provides the possibility to formulate the boundary conditions in a natural way. Together with the use of a direct matrix solver this gives a convergence speed-up of more than a factor of 100 when compared to a scalar formulation on the azimuthal magnetic field that uses an iterative solver. As a result this electromagnetic model is ready to act in future studies as part of the self-consistent description of plasma-electromagnetic coupling. With the electromagnetic model three types of configuration were studied: the closed, semi-open and open configurations, all three based on the same simplified model-plasmas. It is found that the closed configuration, acting as a cavity for which the (de)tuning is extremely sensitive for the plasma conditions, is less suitable for applications in which changes in plasma compositions can be expected. The semi-open configuration can be seen as a model for the practice often used in laboratories to place Microwave Induced Plasma torches in a grid that aims to protect the environment against microwave electromagnetic radiation. Calculations show that this is good practice provided the radius of this cylindrical grid is in the order of 90 mm. For the most often used configuration, the open version, we found that the power balance as expressed by the coefficients of absorption, transmission and reflection, depends on the electron density of the plasma. The reason is that the plasma acts as an antenna, that converts the electromagnetic waves from the co-axial structure to that of the expansion region and that this antenna function depends on the electron density. The influence of various other antenna elements is investigated as well.

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“…Note that this boundary condition assumes a plane wave propagating in the positive z-direction E r ∝ exp(−iβz). This boundary condition is described in detail in [38,41,42].…”

Section: Homogeneous Inhomoneous Dirichletmentioning

confidence: 99%

Section: Homogeneous Inhomoneous Dirichletmentioning

confidence: 99%

Effect of remote field electromagnetic boundary conditions on microwave-induced plasma torches

Jimenez-Diaz¹,

Dijk²,

Mullen³

2011

J. Phys. D: Appl. Phys.

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“…To implement absorbing boundary conditions in our simulations, we extend the results in [24][25][26] as described below.…”

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confidence: 99%

Vacuum topology and the electroweak phase transition

Zhang¹,

Ferrer²,

Vachaspati³

2017

Phys. Rev. D

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We investigate if the topology of pure gauge fields in the electroweak vacuum can play a role in classical dynamics at the electroweak phase transition. Our numerical analysis shows that magnetic fields are produced if the initial vacuum has non-trivial Chern-Simons number, and the fields are helical if the Chern-Simons number changes during the phase transition.An explanation for the observed cosmic matterantimatter asymmetry likely requires CP violating particle interactions at energies at or above the electroweak scale, at an epoch when the universe was out of thermal equilibrium [1]. Models of matter-genesis, more specifically, baryogenesis or leptogenesis, also necessarily involve the violation of baryon plus lepton (B+L) number through anomalous quantum processes. Several studies have now shown that the anomalous violation of B+L at the time of electroweak symmetry breaking, when the Higgs (Φ) acquires a non-vanishing vacuum expectation value (VEV), leads to the production of helical magnetic fields [2,3]. The connection of matter-genesis and magneto-genesis offers a means to probe fundamental particle interactions by the observation of magnetic fields in the universe.In hindsight it is not difficult to intuitively understand the production of helical magnetic fields when B+L is violated by anomalous processes. To change B+L, requires a change in the Chern-Simons number of the electroweak gauge fields and, post electroweak symmetry breaking, this requires passage through a "sphaleron" [4] that has the interpretation of a twisted magnetic monopole-antimonopole configuration [5][6][7] The decay of the sphaleron corresponds to the annihilation of the monopole and antimonopole, with the release of helical magnetic fields [8,9].In the present paper, we address a related questioncan the topology of the electroweak vacuum play a role in the dynamics of the electroweak phase transition? A hint that the answer is in the affirmative is suggested by the work of Jackiw and Pi [10], where they consider a pure vacuum SU(2) gauge field configuration that has non-vanishing Chern-Simons number. They then project the gauge field configuration onto a fixed isospin direction to simulate the effects of the Higgs field VEV, and then evolve and calculate the helicity in the electromagnetic (EM) field. Jackiw and Pi find a non-vanishing EM helicity and further provide the neat result that the EM helicity at late times is 1/2 of the helicity at early times.As originally discussed in Ref.[10], the Jackiw-Pi result depends crucially on their model for projection of the gauge fields in isospin space. For example, note that the initial gauge configuration is pure gauge and has zero energy, while the final configuration with helical magnetic FIG. 1. In a first-order electroweak phase transition, bubbles of true vacuum (|Φ| = η) will grow and encapsulate regions of false vacuum (Φ = 0), within which gauge fields with localized non-trivial Chern-Simons number, NCS, may exist. With time, the bubbles of true vacuum will grow and complete t...

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“…Conditions similar to Equation (7) were introduced for a twodimensional wave equation [11] (first-order absorbing boundary conditions) and used on artificial boundaries for wave-like equations [12][13][14].…”

Section: Rementioning

confidence: 99%